CHAP. XVIH] MISCELLANEOUS RESULTS ON PfilMES. 437 E. Dormoy284 noted that, if 2, 3, . . . , r, s, t, u are the primes in natural order, all primes (and no others) <u2 are given by 2-3 . . .8tm+Dflt+tCiPtfii+t8CiC9D1ar+ ... . . .5-3CtC8. . .C3, where Ct is found from the quotients obtained in finding the g. c. d. ,pf t and 2-3 . . .rs by a rule which if applied to four quotients a, 6,,c, d consists in forming in turn 1, p = dc+lt pb+d, (pb+d)a+p = Ct. Further, Dt = tCt^l} the sign being + or according as there is an odd or an even number of operations in the g. c. d. process. C. de Polignac284" wrote pn for the nth prime and discussed the express-ibility of all numbers, under a specified limit and divisible by no one of Pi,..., Pn-i, in the form (P2, P3, , Pn-l, Pn) + (?3, Z>4, - - , Pn Pi) + - + (?1, . . - , Pn-l) , where (a, 6, . . . ) denotes =*= a°6/3 .... For example, every number < 53 and divisible by neither 2 nor 3 is of the form =fc3a=±=2/3. J. J. Sylvester285 proved that if m is prime to i and not less than n, the product (m+i)(m+2i) . . . (ra+m) is divisible by some prune >n. A. A. Markow286 found a fragment in a manuscript by Tchebychef aiding him to prove the latter Js result that if ^ is the greatest prime divisor of (1+22) (1+42) . . . (1+4AT2), then »/N increases without limit with N (cf. Hermite, Cours, ed. 4, 1891, 197). J. Iwanow287 generalized the preceding theorem as follows: If /x is the greatest prime divisor of (A+l2). . .(A+L2), then /x/L increases without limit with L. C. Stormer288 concluded the existence of an infinitude of primes from Tcheby chef's286 result and used the latter to prove that i(i 1) (i 2) . . . (i ri) is neither real nor purely imaginary if n is any integer 7^3, and i \/ 1. E. Landau47 (pp. 559-564) discussed the topics in the last three papers. Braun130 proved that the (n+l)th prime is the only root XT± 1 of where ai = 2, a2, . . . , an are the first n primes. C. Isenkrahe289 expressed a prime in terms of the preceding primes. R. Le Vavasseur290 noted that all primes between pn and pzn+i, where pn is the nth prime, are given by SJI" #tW»Pn/pt (mod Pn), where Pn = pip2 - pn and WiPJpi=l (mod p,-). 2MComptes Rendus Paris, 63, 1866, 178-181. 284flComptes Rendus Paris, 104, 1887, 1688-90. "'Messenger Math., 21, 1891-2, 1-19, 192. Math. Quest. Educ. Times, 56, 1892, 25. 286Bull. Acad. Sc. St. Pe"tersbourg, 3, 1895, 55-8. 287/6id., 361-6. 388Archiv Math, og Natur., Kristiania, 24, 1901-2, No. 5. J88Math. Annalen, 53, 1900, 42. . Ac. Sc. Toulouse, (10), 3, 1903, 36-8.